Properties

Degree 1
Conductor 569
Sign $0.999 + 0.0406i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.999 + 0.0442i)2-s + (−0.883 + 0.467i)3-s + (0.996 − 0.0883i)4-s + (0.903 + 0.428i)5-s + (0.862 − 0.506i)6-s + (0.325 − 0.945i)7-s + (−0.991 + 0.132i)8-s + (0.562 − 0.826i)9-s + (−0.921 − 0.387i)10-s + (−0.730 + 0.683i)11-s + (−0.839 + 0.544i)12-s + (0.937 + 0.346i)13-s + (−0.283 + 0.958i)14-s + (−0.999 + 0.0442i)15-s + (0.984 − 0.176i)16-s + (−0.448 − 0.894i)17-s + ⋯
L(s,χ)  = 1  + (−0.999 + 0.0442i)2-s + (−0.883 + 0.467i)3-s + (0.996 − 0.0883i)4-s + (0.903 + 0.428i)5-s + (0.862 − 0.506i)6-s + (0.325 − 0.945i)7-s + (−0.991 + 0.132i)8-s + (0.562 − 0.826i)9-s + (−0.921 − 0.387i)10-s + (−0.730 + 0.683i)11-s + (−0.839 + 0.544i)12-s + (0.937 + 0.346i)13-s + (−0.283 + 0.958i)14-s + (−0.999 + 0.0442i)15-s + (0.984 − 0.176i)16-s + (−0.448 − 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.0406i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.0406i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.999 + 0.0406i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (90, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.999 + 0.0406i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8175198352 + 0.01663302860i$
$L(\frac12,\chi)$  $\approx$  $0.8175198352 + 0.01663302860i$
$L(\chi,1)$  $\approx$  0.6855156469 + 0.05801844125i
$L(1,\chi)$  $\approx$  0.6855156469 + 0.05801844125i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−23.53394320554986334576147099408, −22.214247529206124324402866790025, −21.403854894810584729129548039951, −20.94697223044142394704832837116, −19.714645717417994225239606554245, −18.64984137029716909069102552080, −18.22845381103752477805911999585, −17.55300643878439263336819877731, −16.798898792767955654670505371538, −15.89919904618912243860140783110, −15.25715790450623900996157385300, −13.56302889834858857758836174945, −12.99680962567732236724552185414, −11.85256096309141388496587089120, −11.24261252214392200946642171666, −10.33937471851751730822469835314, −9.443934355462773058342266222503, −8.39480417701599606346385129512, −7.79854337306656993186043288413, −6.21229037201043475270076670267, −5.97737690296299458680174685045, −5.006490414737415847493299174651, −2.983792643570901574927212509945, −1.86479214244051369644887187624, −1.046797893936335438250812859649, 0.83769814314544155061383436617, 1.97899223323403466795455662340, 3.33755874240185437977830963026, 4.78044350119525238501203753166, 5.71213485844162385008340871084, 6.82407371929972757800346514629, 7.228503964801459106050167399190, 8.70315031051249981298687110028, 9.71081544798197849660469185263, 10.331784282322417956461891394594, 10.9311709943538584984354824689, 11.72678005079091513120419153177, 12.981103895389424854693578031377, 14.04129091236408505495234185496, 15.06459289419372252160268183807, 16.09953692419794620659391581757, 16.586982824257810889096924975094, 17.61803755359629586028415346363, 18.05323893162973818927475879569, 18.59569561768644502306713259230, 20.18085000785180241473342157895, 20.74318995799991477603820954579, 21.32415656099398634259867926531, 22.53183376455924634619410149489, 23.20473255300451491131874589557

Graph of the $Z$-function along the critical line